• 


11 a bus  of  a  course  In 
^ne  An'O.ytlc 
Geometry . 

by 
-erly 


UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


SYLLABUS 


OF   A 


COURSE  IN  PLANE  ANALYTIC  GEOMETRY. 


INTRODUCTORY. 


BOSTON: 

PUBLISHED  BY  GINN,  HEATH,  &  CO. 
1884. 

COPYRIGHTED  BY  GINN,  HEATH,  &  Co.,  1888. 


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STLLABUS 


INTEODUCTOEY  COUESE  IN  PLANE  ANALYTIC 
GEOIETEY, 


1.  Describe  the  object  of  Analytic  Geometry.     Show  that, 
on  account  of   the   inherent   difference  between  Algebra  and 
Geometry,  two  important  difficulties  present  themselves  at  the 
outset  in  our  attempt  to  apply  Algebra  to  Geometiy,  the  geo- 
metrical interpretation  of  the  distinction  between  positive  and 
negative  quantities,  and  the  algebraic  representation  of  position. 

2.  Show  that  the  first  difficulty  has  been  already  met  and 
overcome  in  Trigonometry.     The  rule  that  opposite  signs  are 
interpreted  geometrically  by  opposite   directions  is   adopted   in 
Analytic  Geometry.     Illustrations. 

[1]     AB  =  -BA. 

3.  Show  that  the  position  of  a  point  is  known  when  we  know 
its  distances  from  two  given  intersecting  lines. 

Explain  rectangular  coordinates.  Define  axes,  origin,  abscissa, 
ordinate,  coordinates,  axis  of  abscissas,  axis  of  ordinates. 

Give  the  rule  concerning  the  signs  of  the  coordinates  of  a 
point. 

Explain  the  different  ways  of  writing  the  coordinates  of  a 
point. 


462617 


4.  It  is  necessary  to  devise  means  of  performing  algebraically 
the  various  operations  required  in  Geometry.  We  begin  with 
the  length  of  a  given  line. 

Obtain  a  formula  for  the  distance  between  two  given  points 
/  and  ajj,  2 . 


[2]     D  = 

Show,  by  considering  different  positions  of  the  points,  that 
the  formula  [2]  is  entirely  general.  Examples.  Geometrical 
problems. 

5.    Find  the  coordinates  of  a  point  bisecting  a  given  line. 


1  J  2 

Find  the  coordinates  of  a  point  dividing  a  given  line  in  a 
given  ratio  mx :  m2. 

_  m2  Xi  -f  mi  x2  _  m2  yi  +  ml  ?/2 

m2  +  OTI    '  m2  +  mi 

Examples.     Geometrical  problems. 

6.  Define  what  is  meant  in  Geometry  by  a  locus.  Examples 
and  problems  in  loci. 

If  the  given  conditions  can  be  expressed  by  an  equation  be- 
tween the  coordinates  of  a  point,  the  curve  generated  by  a 
point  so  moving  that  its  coordinates  always  satisfy  the  equation 
is  called  the  locus  of  the  equation.  Examples.  Numerous  prob- 
lems in  forming  equations  of  simple  loci. 

Give  a  second  definition  of  the  locus  of  an  equation.  Show 
that  every  equation  between  x  and  y  has  a  locus. 

Explain  the  method  of  constructing  the  locus  of  an  equation 
by  finding  points  of  the  required  curve.  Examples. 

Define  the  intercepts  of  a  curve  on  the  axes,  and  show  how 
they  may  be  found  from  the  equation. 

Explain  the  method  of  finding  the  points  of  intersection  of 
two  curves  whose  equations  are  given.  Examples. 


THE  STRAIGHT  LINE. 

7.    Find  the  equation  of  a  line  when  the  intercepts  on  the 
axes  are  given. 


Show  that  [5]  is  correct  wherever  the  point  (#,?/)  may  be 
taken  on  the  line,  and  whatever  the  signs  of  a  and  b. 

Find  the  equation  of  a  line  when  two  points  of  the  line  are 
given. 

[6]    x~xi  _.  y—y* 

When  the  intercept  b  on  the  axis  of  ordinates  and  the  angle  y 
made  with  the  axis  of  abscissas  are  given, 

[7]     y  =  lx-\-b,     where  1  =  tan  y. 

When  the  coordinates  of  a  point  through  which  the  line 
passes  and  the  angle  made  with  the  axis  of  abscissas  are  given, 

[8]     y  —  7/1  =  l(x  —  XT)  . 
8.    Show  that  every  equation  of  the  first  degree 


is  the  equation  of  a  straight  line,  and  prove  that 

TO  „— *    *=-£,  «=_£ 


9.  Show  how  to  obtain  the  equation  of  a  line  through  two 
given  points  by  determining  the  coefficients  of  Ax+By-\-C =  0? 
so  that  the  equation  shall  be  satisfied  by  the  given  coordinates. 


10.  Find  from  a  geometrical  figure  the  condition  that  two 
lines  whose  equations  are  given  shall  be  parallel, 

[10]     Z1  =  Z,         or,     |j  =  |; 
that  they  shall  be  perpendicular, 


[U]    ^-1,    or,    |;  =  -f 


Problems. 


11.    Find  the  angle  between  two  straight  lines  whose  equa- 
tions are  given. 

+™ a  — 
- 


Obtain  formulas  [10]  and  [11]  from  [12]. 

12.  Explain  the  method  of  finding  the  equation  of  a  line 
passing  through  a  given  point  and  parallel  to  a  given  line  ; 
perpendicular  to  a  given  line.     Use  undetermined  coefficients. 
Geometrical  problems. 

13.  Devise  a  method  for  finding  the  distance  from  a  given 
point  to  a  given  line.     Obtain  by  it  the  formula 

[13]     D  = 


14.    Find  the  area  of  a  triangle  when  the  coordinates  of  its 
vertices  are  given. 

[14]     M= 


15.    Describe  the  general  method  of  finding  the  equation  of 
any  given  geometrical  locus.     Numerous  problems. 


TRANSFORMATION  OF  COORDINATES. 

16.  Obtain  formulas  for  transforming  from  one  system  of 
coordinates  to   another  when   the   new   axes  are  respectively 
parallel  to  the  old. 

[15]      x  =  X0-\-x'',        y  =  y0  +  y'. 

17.  Obtain  formulas  for  transforming  from  one  system  of 
coordinates  to  another  having  the  same  origin. 


[16]     a;  =  ;c'cos0  —  y' siu6  ;      y  =  x'sinO-\-  y' cosO. 

18.    Explain  polar  coordinates.     Obtain  formulas  for  trans- 
forming from  rectangular  to  polar  coordinates. 


[17]     x  = 
Problems  in  polar  coordinates. 


THE  CIRCLE. 
19.   Find  the  equation  of  the  circle  in  the  form 

[18]     (x-a)*  +  (y-b)*  =  i*. 
When  the  centre  is  at  the  origin,  this  becomes 

[19] 


20.    Show,  by  expanding  [18],  that  the  equation  of  any  circle 
may  be  written  in  the  form 

[20]     x2  +  y2+Dx+Ey+F=0. 

Explain  the  method  of  finding  the  centre  and  radius  of  a 
circle  whose  equation  is  given  in  expanded  form. 


21.  Give  two  methods  of  finding  the  equation  of  a  circle 
passing  through  three  given  points. 

22.  Find  the  equation  of  a  tangent  to  the  circle  [19]  at  a 
given  point  on  the  circumference. 

[21] 


23.  Define  a  normal  to  a  curve.     Find  the  equation  of  the 
normal  at  a  given  point  of  the  circle.     [19]. 

f~OO~l  f\ 

L     J     2/1  *^  ~"~  *^i  2/ ""~"    * 

24.  Find  the  locus  of   the  middle  points   of   a   system  of 
parallel  chords. 


[23]     a  +  tan0.?/=  0. 

Such  a  line  is  called  a  diameter. 

Prove  that  every  chord  through  the  centre  is  a  diameter. 

Problems  on  the  circle. 


THE  COKIC  SECTIONS. 

25.    Define  the  ellipse,  the  hyperbola,  the  parabola,  and  find 
their  equations  in  the  forms 

[24]     -  +  ^=i        —  _tf=\        y2=2mx. 
1     J     a?^W        '       a?      W 

Prove   that  in   the   ellipse  and   the   hyperbola   every  chord 
through  the  centre  is  bisected  by  the  centre. 


26.    Find  the  equations  of  the  tangent  and  normal  to  each  of 
these  curves.     Prove  that  the  tangents  at  the  opposite  extremi- 


ties  of  a  chord  through  the  centre  in  the  ellipse  and  the  hyper- 
bola are  parallel. 

[25]     Tangents.    £?  +  M=i,  ^-M=i, 

a-        b2  a-        V 


a2        b2  a2        b2 

[26]     Normals,     —x  --  y  =  a2—b2,  —x-\  —  y  =  a2-f-62, 

xi      y\  xi      y\ 
y\ 


27.  Prove  that  the  tangent  and  normal  at  any  point  of  the 
ellipse  or  the  hyperbola  bisect  the  angle  between  the  focal  radii 
drawn  to  the  point. 

Show  how  this  property  can  be  used  in  drawing  a  tangent  to 
the  curve. 

Prove  that  con-focal  conies  intersect  at  right  angles. 


28.  Prove  that  if  on  the  major  axis  of  an  ellipse  as  a  diam- 
eter a  circle  be  described,  the  ordinate  of  any  point  of  the 
ellipse  will  be  to  the  ordinate  of  the  point  of  the  circle  having 
the  same  abscissa  as  b :  a. 

Prove  that  the  area  of  an  ellipse  is  -n-ab. 


29.  Define  the  as3*mp  totes  of  an  hyperbola  and  find  then* 
equations.  Prove  that  a  point  moving  along  the  hyperbola 
away  from  the  centre  approaches  indefinitely  near  to  the  asymp- 
tote but  never  reaches  it. 


30.  Prove  that  the  angle  between  the  focal  radius  of  a  point 
on  the  parabola  and  a  line  through  the  point  parallel  to  the 
principal  axis  is  bisected  by  the  normal  at  the  point. 


462G17 


8 

31.   Find  for  the  ellipse  the  equation  of  the  diameter  bisecting 
a  given  set  of  parallel  chords  ;  for  the  hyperbola. 


[27]     Z/2z  +  a2tan0.2/=0,          b*x  —  a*tan0.?/  =  0. 
Prove  that  every  chord  through  the  centre  is  a  diameter. 

32.  Find  the  equation  of  the  diameter  bisecting  a  given  set 
of  parallel  chords  in  the  parabola. 

[28]     tau0.y  =  m. 

33.  Describe  the  method  of  finding  the  centre  of  an  ellipse 
or  an  hyperbola  when  an  arc  of  the  curve  is  given. 

34.  Show  how  to  find  out  whether  a  given  arc  known  to 
belong  to  a  conic  section  is  part  of  an  ellipse,  of  an  hyperbola, 
or  of  a  parabola. 

35.  Prove  that  if  of  two  diameters  in  a  central  conic  the  first 
bisects  the  chords  parallel  to  the  second,  the  second  will  bisect 
the  chords  parallel  to  the  first.     Such  a  pair  of  diameters  are 
called  conjugate. 

Find  the  equation  of  the  diameter  conjugate  to  the  diameter 
through  a  given  point  (a^,  2/1)  of  an  ellipse. 


Show  that  the  tangent  at  a  given  point  of  a  central  conic  can 
be  drawn  by  the  aid  of  a  pair  of  conjugate  diameters. 

36.    Find  polar  equations  for  the  circle,  the  ellipse,  the  hyper- 
bola, and  the  parabola. 

W.  E.  BYERLY, 

Professor  of  Mathematics  in  Harvard  University. 


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552    Byerly  - 
B98sy  Syllabus  of  a 
1884   course  in  plane 

analytic 
geometry . 


552 

B98sy 

1884 


